Abstract
In this paper, we define a finite sum analogue of multiple polylogarithms inspired by the work of Kaneko and Zagier (in the article “Finite multiple zeta values” in preparation) and prove that they satisfy a certain analogue of the shuffle relation. Our result is obtained by using a certain partial fraction decomposition which is an idea due to Komori et al. (Math Z 268:993–1011, 2011). As a corollary, we give an algebraic interpretation of our shuffle product.
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Borwein, M.J., Bradley, M.D., Broadhurst, D.J., Lisoněk, P.: Special values of multiple polylogarithms. Trans. Am. Math. Soc. 353(3), 907–941 (2001)
Besser, A.: Finite and \(p\)-adic polylogarithms. Compos. Math. 130, 215–223 (2002)
Broadhurst, D.J., Kreimer, D.: Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B 393, 403–412 (1997)
Brown, F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. Math. 175(2), 949–976 (2012)
Deligne, P., Goncharov, A.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. 38(4), 1–56 (2005)
Drinfel’d, V.G.: On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(\(\overline{\mathbb{Q}}/{\mathbb{Q}}\)). Leningrad Math. J. 2(4), 829–860 (1991)
Elbaz-Vincent, P., Gangl, H.: On poly(ana)logs I. Compos. Math. 130, 161–214 (2002)
Goncharov, A.B.: Multiple polylogarithms, cyclotomy and modular complexes. Math. Res. Lett. 5(4), 497–516 (1998)
Komori, Y., Matsumoto, K., Tsumura, H.: Shuffle products of multiple zeta values and partial fraction decompositions of zeta-functions of root systems. Math. Z. 268, 993–1011 (2011)
Kontsevich, M.: The 1+1/2 logarithm, appendix to [EG]. Compos. Math. 130, 211–214 (2002)
Le, T.Q.T., Murakami, J.: Kontsevich’s integral for the Homfly polynomial and relations between values of the multiple zeta functions. Topol. Appl. 62, 193–206 (1995)
Minh, H.N., Petitot, M., Van Der Hoeven, J.: Shuffle algebra and polylogarithms. Discrete Math. 225, 217–230 (2000)
Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de l’unité. Publ. Math. IHES 95, 185–231 (2002)
Terasoma, T.: Mixed Tate motives and multiple zeta values. Invent. Math. 149, 339–369 (2002)
Weil, A.: Elliptic Functions According to Eisenstein and Kronecker. Springer, Berlin, Heidelberg, New York (1976)
Acknowledgments
The authors would like to thank the members of the KiPAS-AGNT group for giving us a great environment to study and reading the manuscript carefully, and members of the Department of Mathematics at Keio University for their hospitality. This research was supported in part by JSPS KAKENHI 21674001, 26247004.
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Ono, M., Yamamoto, S. Shuffle product of finite multiple polylogarithms. manuscripta math. 152, 153–166 (2017). https://doi.org/10.1007/s00229-016-0856-9
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DOI: https://doi.org/10.1007/s00229-016-0856-9